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Unformatted text preview: ECE 562 Fall 2011 PRACTICE EXAM 1 Solutions 1. (15 pts total) True or False. Determine if the following statements are True or False. You need to provide a brief justification for your answer to get credit. (a) The complex baseband representation of √ 2 cos( πt ) cos(2 πf c t ) is cos( πt ). Ans: True. By the definition of the complex baseband representation. (b) If X = X I + jX Q is a proper complex random variable then Var( X I + X Q ) = Var( X I ) + Var( X Q ) Ans: True. Since X I and X Q are uncorrelated. (c) The maximum likelihood detector always minimizes the error probability. Ans: False. Only if the symbols are equally likely. (d) For PSK modulation on an AWGN channel, the Intelligent Union Bound and Nearest Neighbor Approximation always coincide. Ans: True. The nearest neighbors cover the entire error region. (e) For 4-ary orthogonal modulation, P b = P e / 2. Ans: False. P b = 2 3 P e 2. (11 pts total) Gram-Schmidt. Consider the following signal set. t s 3 ( t ) s 1 ( t ) s 2 ( t ) 1 2 t 1 1 t − 1 1 2 t − 1 1 2 1 s 4 ( t ) (a) (4 pts) Find an orthonormal basis for the span of this signal set. (Think before you answer since the G-S solution is not unique.)you answer since the G-S solution is not unique....
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